Measures of Pseudorandomness for Finite Sequences: Minimal Values

Abstract

Mauduit and Sárközy introduced and studied certain numerical parameters associated to finite binary sequences $E_N\in\{-1,1\}^N$ in order to measure their ‘level of randomness’. Two of these parameters are the normality measure$\cal{N}(E_N)$ and the correlation measure$C_k(E_N)$of order k, which focus on different combinatorial aspects of $E_N$. In their work, amongst others, Mauduit and Sárközy investigated the minimal possible value of these parameters.

In this paper, we continue the work in this direction and prove a lower bound for the correlation measure $C_k(E_N)$ (k even) for arbitrary sequences $E_N$, establishing one of their conjectures. We also give an algebraic construction for a sequence $E_N$ with small normality measure $\cal{N}(E_N)$.